csat-suneung 2006 Q9

csat-suneung · South-Korea · csat__math-science 3 marks Stationary points and optimisation Analyze function behavior from graph or table of derivative
The function $y = f ( x )$ is continuous on all real numbers, and for all $x$ with $| x | \neq 1$, the derivative $f ^ { \prime } ( x )$ is
$$f ^ { \prime } ( x ) = \left\{ \begin{array} { c c } x ^ { 2 } & ( | x | < 1 ) \\ - 1 & ( | x | > 1 ) \end{array} \right.$$
Which of the following statements in are true? [3 points]

ㄱ. The function $y = f ( x )$ has an extremum at $x = - 1$. ㄴ. For all real numbers $x$, $f ( x ) = f ( - x )$. ㄷ. If $f ( 0 ) = 0$, then $f ( 1 ) > 0$.
(1) ㄱ
(2) ㄴ
(3) ㄷ
(4) ㄱ, ㄷ
(5) ㄱ, ㄴ, ㄷ
The function $y = f ( x )$ is continuous on all real numbers, and for all $x$ with $| x | \neq 1$, the derivative $f ^ { \prime } ( x )$ is

$$f ^ { \prime } ( x ) = \left\{ \begin{array} { c c } 
x ^ { 2 } & ( | x | < 1 ) \\
- 1 & ( | x | > 1 )
\end{array} \right.$$

Which of the following statements in <Remarks> are true? [3 points]

<Remarks>

\noindent ㄱ. The function $y = f ( x )$ has an extremum at $x = - 1$.\\
ㄴ. For all real numbers $x$, $f ( x ) = f ( - x )$.\\
ㄷ. If $f ( 0 ) = 0$, then $f ( 1 ) > 0$.\\
(1) ㄱ\\
(2) ㄴ\\
(3) ㄷ\\
(4) ㄱ, ㄷ\\
(5) ㄱ, ㄴ, ㄷ